Re Pretty "gnomonic" recurrence

[N. J. A. Sloane]

That array is of course already in the database:

%I A048004
%S A048004 1,1,1,1,2,1,1,4,2,1,1,7,5,2,1,1,12,11,5,2,1,1,20,23,12,5,2,1,1,33,47,
%T A048004 27,12,5,2,1,1,54,94,59,28,12,5,2,1,1,88,185,127,63,28,12,5,2,1,1,143,
%U A048004 360,269,139,64,28,12,5,2,1,1,232,694,563,303
%N A048004 Array T by rows: T(h,k)=number of zero-one words of length h and maximal runlength of 0 equal to k.
%O A048004 1,5
%K A048004 nonn,tabl
%A A048004 Clark Kimberling, ck6 at cedar.evansville.edu
%Y A048004 T(h,2)=a(h+1) for h=2,3,..., where a=A000071.
%Y A048004 T(h,3)=b(h) for h=3,4,..., where b=A000100.
%Y A048004 T(h,4)=c(h) for h=4,5,..., where c=A000102.
%Y A048004 Cf. A048003.
%e A048004 Rows: {1}; {1,1}; {1,2,1}; {1,4,2,1}; ...


i will add at least one of your recuurences to the entry - NJAS